*[Added another complementary question below.]*

## Motivation

The 1-skeleton of the triangular bipyramid seems to be the smallest connected planar graph $G$ with the following

Property:There is a cycle $\gamma$ ($\color{red}{\mathsf{red}}$) in $G$ with exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ ($\color{blue}{\mathsf{blue}}$) is contained in theinteriorof $\pi(\gamma)$ and the other component $G_2$ ($\color{black}{\mathsf{black}}$) is contained in theexteriorof $\pi(\gamma)$ – or vice versa.

Definition: A cycle $\gamma$ in the graph $G$ is aif $G - \gamma$ splits up into exactly two connected components $G_1, G_2$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ is contained in the exterior of $\pi(\gamma)$ – or vice versa.Jordan cycle

## Questions

(How) can the property of

being a Jordan cycle$\gamma$ be defined purely combinatorial, without mentioning graph embeddings $\pi$ and Jordan curves $\pi(\gamma)$?(How) can planar graphs

containing a Jordan cyclebe characterized purely combinatorially?

ADDED: Can planar graphs be characterized in whicheverycycle is a Jordan cycle?